Algebra Examples

$f (x) = 2 x^{3} + 7 x^{2} + 98 x + 343$

Step 1

Factor out the greatest common factor from each group.

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Step 1.1

Group the first two terms and the last two terms.

$f (x) = (2 x^{3} + 7 x^{2}) + 98 x + 343$

Step 1.2

Factor out the greatest common factor (GCF) from each group.

$f (x) = x^{2} (2 x + 7) + 49 (2 x + 7)$

Step 2

Factor the polynomial by factoring out the greatest common factor, $2 x + 7$ .

$f (x) = (2 x + 7) (x^{2} + 49)$

Step 3

Factor over the complex numbers.

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Step 3.1

Multiply the constant in the polynomial $x^{2} + 49$ by $- i^{2}$ where $i^{2}$ is equal to $- 1$ .

$f (x) = (2 x + 7) (x^{2} - 49 i^{2})$

Step 3.2

Rewrite $49 i^{2}$ as ${(7 i)}^{2}$ .

$f (x) = (2 x + 7) (x^{2} - {(7 i)}^{2})$

Step 3.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - i^{2} = (a + i) (a - i)$ where $a = x$ and $i = 7 i$ .

$f (x) = (2 x + 7) ((x + 7 i) (x - (7 i)))$

Step 3.4

Factor.

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Step 3.4.1

Multiply $7$ by $- 1$ .

$f (x) = (2 x + 7) ((x + 7 i) (x - 7 i))$

Step 3.4.2

Remove unnecessary parentheses.

$f (x) = (2 x + 7) (x + 7 i) (x - 7 i)$